1. What is Modulo Operation?

The modulo operation finds the remainder after division of one number by another. Given two positive numbers, a (dividend) and b (divisor), a modulo b is the remainder of the Euclidean division of a by b.

2. How Does Modulo Work?

The modulo operation follows this mathematical formula:

Where:

• \( a \) — Dividend (the number being divided)
• \( b \) — Divisor (the number dividing by)
• \( q \) — Quotient (the integer result of division)
• \( r \) — Remainder (the result of modulo operation)

Example: 17 mod 5 = 2 because 17 ÷ 5 = 3 with remainder 2.

3. Practical Applications

Details: Modulo operations are used in programming, cryptography, time calculations, and many mathematical algorithms. It's essential for determining even/odd numbers, wrapping values, and cyclic operations.

4. Using the Calculator

Tips: Enter the dividend (number to be divided) and divisor (number to divide by). The divisor must be non-zero. The calculator will show the remainder after division.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between modulo and remainder?
A: For positive numbers they're the same, but they differ with negative numbers. Modulo always returns a positive result.

Q2: Can the divisor be zero?
A: No, division by zero is undefined in mathematics, so modulo by zero is also undefined.

Q3: How is modulo used in programming?
A: Common uses include array indexing, hash functions, checking for even/odd, and implementing circular buffers.

Q4: What about negative numbers?
A: This calculator handles negative numbers correctly according to mathematical definition (result is always non-negative).

Q5: Is modulo the same as division?
A: No, division gives the quotient while modulo gives the remainder after division.